276 X EXISTENCE THEOREMS

(a) Suppose there exists a sequence (un) of analytic mappings of A info B such that
UQ(X) = 0 in A and un(x)= U(x, un~i(*))' x in A for n ^ 1,. Suppose in addition
that for every compact subset L of A, the restrictions of the un to L form a relatively
compact subset of #E(L). Prove that the sequence (un) converges uniformly in any
compact subset of A to an analytic mapping v of A into B such that v(x) — U(x, v(x)) - x
in A; furthermore, v is the unique mapping satisfying that equation (use (10.2.1) and
(9.13.2)).

(b) Suppose that in E, A and B are the open balls of center 0 and radii a and b.
Let <p be a continuous mapping of [0, a[ x [0, b[ into R such that 77-^(£,77) is
increasing in [0,6[ for every f e [Q,a] and suppose that \\U(x,y)\\ ^ 9(||*||, \\y\\) in
A x B. Suppose in addition that there exists a continuous mapping 6 of [0, a[ into
[0, b[ such that 0(f) = <p(£, #(£))£ in [0, a]. Prove that under these conditions there
is a unique analytic mapping v of A into B such that v(x) = U(x, v(x)) • x in A, and
that |!ğ(jc)||^ 0(ll*||) in A (use (a); prove the existence of the mappings un by induction
on /?).

(c) Suppose A and B are defined as in (b): let ^(77) be the l.u.b. of \\U(x,y)\\ for
jjjc|| < a, \\y\\ < 77, when 77 > 0, and take 0(0) = 0(0+). Suppose that 0(0) >0
and that the function 77 -> 77/0(77) is increasing in some interval [0, y[, where y ^ 6,
and y/0(y—) =^ a. Then there is a unique analytic mapping v of the open ball P of
center 0 and radius y/0(y—) into B, such that v(x) = U(x, v(x)) • x in P.

10. Let /, g be two complex valued analytic functions defined in a neighborhood of the
closed polydisk P c c2 of center (0,0) and radii a, b. Let M (resp. N) be the l.u.b. of
|/(x,y)| (resp. \g(x9 y)\) for \x\ = a and |j| ^ b (resp. for \x\ ^ a and \y\ = b). Then,
there exist two uniquely determined functions u(s9 /), v(s, /), analytic for \s\ < a/M
and \t\ < &/N, such that (u(s9 /), v(s9 /)) e P for (s, t) in the polydisk Q defined by the
previous inequalities and that

u(s, t) — sf(u(s, /), v(s, /)) = 0 and v(st t) — tg(u(s, t), v(s, t)) = 0
in Q. Furthermore, let

1 — s— — s —

<fc %

dx 8y

and let h(x, y, s, t) be an arbitrary analytic function in P x Q; show that
h(u(s, t), v(s, t), s, t) ^

for (s, t) E Q, where cmn is the value for x — y — 0 of the function

I Qm + n

mlniexm8yğL'-"'-- ' —

and the series on the right-hand side is convergent in Q; note that cmn depends on s
and / if h does. ("Lagrange's inversion formula." First apply Rouche's theorem
(9.17.3) to x — sf(x, y), considered as a function of x; this defines an analytic function
w(s, y) such that w(s, y) — sf(w(sty), y) = 0,by (10.2.4); next apply similarly Rouche's
theorem to y — tg(w(s, y), y) considered as a function of y. Finally, let y, S be the
circuits 6~*aeie, 6-^beie in C (0 =^ 6 ^ ITT). Consider the repeated integrally